Simulation of singlet correlation without any communication, but using a weaker ressource: a "non-local machine".

Abstract

The importance of quantum entanglement is by now widely appreciated as a resource for quantum information applications. A unit of entanglement has been identifies and named e-bit; it consists of a pair of maximally entangled qubits, e.g. of a singlet: the same singlet that Bohm used in his version of the EPR paradox. A few years ago connection with communication complexity started to be studied, with question like how much communication is required to simulate an e-bit? From Bell inequality we know that it is impossible to simulate a singlet without any communication even if one assumes that both parties share local hidden variables, or in modern terminology, share randomness. Recently, Tonner and Bacon proved that actually a single bit of communication suffice for perfect simulation.
Independently from the above story, Popescu and Rochlich raised the following question: can there be correlation stronger than the quantum mechanical ones that do not allow one to signal? They answered by showing a hypothetical non-local machine that does not allow signaling, yet violates the CHSH-bell inequality by the absolute maximal value of 4 (while quantum correlation achieve at most $2\sqrt{2}$. They concluded asking why Nature is non-local, but not maximally non-local, where the maximum would be only limited by the no-signaling constraint?
It is straightforward to simulate the PR machine with a single bit of communication. Consequently, the PR nonlocal machine is a strictly weaker resource than a bit of communication. We show that singlets can be simulated using only one instance of the PR non-local machine. Hence, assuming that Nature is sparing with resources, one is be tempted to conclude that she is using something like the non-local machine.
Finally, we raise the question whether correlations arising from partially entangled qubits can be simulated using only an e-bit?